Existence of a Closed Geodesic on Non-compact Riemannian Manifolds with Boundary

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Germinario ◽  
Miguel Sánchez
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesic ◽  
Detailed Comparison ◽  
Manifolds With Boundary ◽  
Manifold With Boundary

AbstractA new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

EXTREMAL FUNCTIONS FOR MOSER–TRUDINGER INEQUALITIES ON 2-DIMENSIONAL COMPACT RIEMANNIAN MANIFOLDS WITH BOUNDARY

2006 ◽  
Vol 17 (03) ◽  
pp. 313-330 ◽  
Author(s):  
YUNYAN YANG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Extremal Functions ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Blowing Up

Let (M,g) be a 2-dimensional compact Riemannian manifold with boundary. In this paper, we use the method of blowing up analysis to prove the existence of the extremal functions for some Moser–Trudinger inequalities on (M,g).

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary

2010 ◽  
Vol 53 (4) ◽  
pp. 674-683 ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Riemannian Manifolds ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Manifolds With Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Manifold With Boundary

AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

scholarly journals A long neck principle for Riemannian spin manifolds with positive scalar curvature

2020 ◽  
Vol 30 (5) ◽  
pp. 1183-1223
Author(s):  
Simone Cecchini
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Pairwise Disjoint ◽  
Index Theory ◽  
Spin Manifold ◽  
Manifolds With Boundary ◽  
Topological Information ◽  
Manifold With Boundary ◽  
Spin Manifolds ◽  
Nonzero Degree

AbstractWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$ scal ( X ) ≥ n ( n - 1 ) and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$ f : X → S n which is strictly area decreasing, then the distance between the support of $$\text {d}f$$ d f and the boundary of X is at most $$\pi /n$$ π / n . This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$ scal ( X ) > σ > 0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $$\partial X$$ ∂ X is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$ π ( n - 1 ) / ( n σ ) . Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$ N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$ scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$ 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.

scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

2019 ◽  
Vol 27 (2) ◽  
pp. 179-211
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Riemannian Manifolds ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Common Boundary

AbstractWe set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C0−metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.

scholarly journals Distance difference functions on nonconvex boundaries of Riemannian manifolds

2021 ◽  
Vol 33 (1) ◽  
pp. 57-64
Author(s):  
S. Ivanov
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Complete Riemannian Manifold ◽  
Manifold With Boundary ◽  
Distance Difference

It is shown that a complete Riemannian manifold with boundary is uniquely determined, up to isometry, by its distance difference representation on the boundary. Unlike previously known results, no restrictions on the boundary are imposed.

scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

2020 ◽  
Vol 28 (1) ◽  
pp. 165-179
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Injectivity Radius ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Convex Boundary

AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

scholarly journals Embedding Topological $n$-Manifolds into Compact Nonstandard Topological $n$-Manifolds with Boundary

2021 ◽  
Author(s):  
Yu-Lin Chou
Keyword(s):  
Topological Manifold ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Topological Manifolds ◽  
One Point Compactification ◽  
Main Message

We show as a main message that there is a simple dimension-preserving way to openly and densely embed every topological manifold into a compact ``nonstandard'' topological manifold with boundary.This class of ``nonstandard'' topological manifolds with boundary contains the usual topological manifolds with boundary.In particular,the Alexandroff one-point compactification of every given topological $n$-manifold is a ``nonstandard'' topological $n$-manifold with boundary.

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